From Riemann Manifolds to Riemann Manifolds - Map Projections: Cartographic Information Systems - page 105

tan l =( Λ 1

Λ 1 + Λ 2 tan 2 Ψ l versus tan r =

tan Ψ l tan Ψ r

λ 1 + λ 2 tan 2 Ψ r

− Λ 2 )

( λ 1 − λ 2 ) .

Box 1.57 (The optimization problem; extremal, left angular shear or right angular shear;
maximal angular distortion).

Optimization problem:

l r =

=

versus (1.336)

{ l ∈ | l ( ψ l )=extr . { r ∈ | r ( ψ r )=extr .

=arg [0 . 2 π ] } =arg [0 . 2 π ] } ,

x := Ψ l ,f ( x ); = l ( Ψ l ) , x := Ψ I ,f ( x ); = r ( Ψ r ) ,

(1.337)

tan x

a := Λ 1 ,b := Λ 2 . tan f ( x )=

a + b tan 2 x .

a := λ 1 , b := λ 2 .

Stationary points:
(tan x ) =1+tan 2 x.

(1.338)

f ( x )=0
⇔
(tan f ( x )) =(1+tan 2 f ( x )) f ( x )=0 ,

1+tan 2 x
( a + b tan 2 x ) 2 ( a

(tan f ( x )) = b tan 2 x ) . −

(tan f ( x )) = 0

(1.339)

⇔

b tan 2 x =0
⇔

a −

a
b ,

tan x = ±

Λ 1
Λ 2

λ 1
λ 2 .

tan Ψ l = versus tan Ψ r =

± ± (1.340)

Extremal left or right angular shear:

tan l versus tan r = ±

1
2

Λ 1 − Λ 2 1

2

λ 1 − λ 2

= ± √ Λ 1 Λ 2 √ λ 1 λ 2 ,